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## G = C162order 162 = 2·34

### Cyclic group

Aliases: C162, also denoted Z162, SmallGroup(162,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C162
 Chief series C1 — C3 — C9 — C27 — C81 — C162
 Lower central C1 — C162
 Upper central C1 — C162

Generators and relations for C162
G = < a | a162=1 >

Smallest permutation representation of C162
Regular action on 162 points
Generators in S162
`(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162)`

`G:=sub<Sym(162)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162)]])`

C162 is a maximal subgroup of   Dic81

162 conjugacy classes

 class 1 2 3A 3B 6A 6B 9A ··· 9F 18A ··· 18F 27A ··· 27R 54A ··· 54R 81A ··· 81BB 162A ··· 162BB order 1 2 3 3 6 6 9 ··· 9 18 ··· 18 27 ··· 27 54 ··· 54 81 ··· 81 162 ··· 162 size 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C6 C9 C18 C27 C54 C81 C162 kernel C162 C81 C54 C27 C18 C9 C6 C3 C2 C1 # reps 1 1 2 2 6 6 18 18 54 54

Matrix representation of C162 in GL1(𝔽163) generated by

 75
`G:=sub<GL(1,GF(163))| [75] >;`

C162 in GAP, Magma, Sage, TeX

`C_{162}`
`% in TeX`

`G:=Group("C162");`
`// GroupNames label`

`G:=SmallGroup(162,2);`
`// by ID`

`G=gap.SmallGroup(162,2);`
`# by ID`

`G:=PCGroup([5,-2,-3,-3,-3,-3,36,57,78]);`
`// Polycyclic`

`G:=Group<a|a^162=1>;`
`// generators/relations`

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